Bacsinszky downed French hope Kristina Mladenovic 6-4, 6-4 on a rain-hit day at Roland Garros, while 19-year-old Ostapenko stunned Caroline Wozniacki 4-6, 6-2, 6-2.
Bacsinszky or Ostapenko can make it a double celebration when they meet on Thursday as both women celebrate their birthdays.
Bacsinszky will be 28; Ostapenko will turn 20.
Tuesday's matches both lasted just under two hours in duration but took the best part of six hours to complete after two rain delays -- one lasting three hours.
The weather interruptions forced Rafael Nadal and Novak Djokovic's quarterfinals to be pushed back until Wednesday, drawing a deafening chorus of boos from the Paris crowd.
